OFFSET
1,2
COMMENTS
It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - Michel Marcus, Feb 28 2019
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.
FORMULA
Multiplicative with a(p^e) = p^2. - Mitch Harris, May 17 2005
G.f.: Sum_{k>=1} mu(k)^2*J_2(k)*x^k/(1 - x^k), where J_2() is the Jordan function. - Ilya Gutkovskiy, Nov 06 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 / 3 = 0.214725... . - Amiram Eldar, Oct 30 2022
a(n) = Sum_{1 <= i, j <= n} ( mobius(n/gcd(i, j, n)) )^2. - Peter Bala, Jan 28 2024
MAPLE
a := n -> mul(f, f=map(x->x^2, select(isprime, divisors(n))));
seq(a(n), n=1..56); # Peter Luschny, Mar 30 2014
MATHEMATICA
a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)
PROG
(PARI) a(n)=my(f=factor(n)[, 1]); prod(i=1, #f, f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Reinhard Zumkeller, Dec 10 2002
STATUS
approved